LMFDB - Embedded newform 8624.2.a.cz.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8624,2,Mod(1,8624)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8624 = 2^{4} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8624.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$68.8629867032$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.89289.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} - x + 15 $ x^4 - 9*x^2 - x + 15
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 616)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.70566$ of defining polynomial
Character	$\chi$	$=$	8624.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.32058 q^{3} +1.70566 q^{5} +8.02623 q^{9} +O(q^{10})$	$q+3.32058 q^{3} +1.70566 q^{5} +8.02623 q^{9} +1.00000 q^{11} -3.32058 q^{13} +5.66376 q^{15} +0.867372 q^{17} -2.27869 q^{19} -2.61492 q^{23} -2.09074 q^{25} +16.6900 q^{27} +9.34681 q^{29} +8.59926 q^{31} +3.32058 q^{33} -7.41131 q^{37} -11.0262 q^{39} +4.83828 q^{41} +6.32058 q^{43} +13.6900 q^{45} +2.40769 q^{47} +2.88018 q^{51} +7.98434 q^{53} +1.70566 q^{55} -7.56655 q^{57} +11.0681 q^{59} -7.43392 q^{61} -5.66376 q^{65} -6.48229 q^{67} -8.68305 q^{69} -0.867372 q^{71} -13.6481 q^{73} -6.94245 q^{75} -9.47581 q^{79} +31.3417 q^{81} -6.54394 q^{83} +1.47944 q^{85} +31.0368 q^{87} +8.86452 q^{89} +28.5545 q^{93} -3.88665 q^{95} +13.2368 q^{97} +8.02623 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - 4 q^{5} + 10 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 - 4 * q^5 + 10 * q^9	$ 4 q + 2 q^{3} - 4 q^{5} + 10 q^{9} + 4 q^{11} - 2 q^{13} + q^{15} + 3 q^{17} + 13 q^{19} - 10 q^{23} + 2 q^{25} + 23 q^{27} + 4 q^{29} + q^{31} + 2 q^{33} - 8 q^{37} - 22 q^{39} + 9 q^{41} + 14 q^{43} + 11 q^{45} + 11 q^{47} + 12 q^{51} - q^{53} - 4 q^{55} - 10 q^{57} + 33 q^{59} - 9 q^{61} - q^{65} - 25 q^{67} - 23 q^{69} - 3 q^{71} + 16 q^{75} - 28 q^{79} + 4 q^{81} - 5 q^{83} - 27 q^{85} + 47 q^{87} + 3 q^{89} + 38 q^{93} - 25 q^{95} + 20 q^{97} + 10 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 - 4 * q^5 + 10 * q^9 + 4 * q^11 - 2 * q^13 + q^15 + 3 * q^17 + 13 * q^19 - 10 * q^23 + 2 * q^25 + 23 * q^27 + 4 * q^29 + q^31 + 2 * q^33 - 8 * q^37 - 22 * q^39 + 9 * q^41 + 14 * q^43 + 11 * q^45 + 11 * q^47 + 12 * q^51 - q^53 - 4 * q^55 - 10 * q^57 + 33 * q^59 - 9 * q^61 - q^65 - 25 * q^67 - 23 * q^69 - 3 * q^71 + 16 * q^75 - 28 * q^79 + 4 * q^81 - 5 * q^83 - 27 * q^85 + 47 * q^87 + 3 * q^89 + 38 * q^93 - 25 * q^95 + 20 * q^97 + 10 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	3.32058	1.91714	0.958568	−	0.284863i	$-0.0919483\pi$
$3$	3.32058	1.91714	0.958568	+	0.284863i	$0.0919483\pi$
$4$	0	0
$5$	1.70566	0.762793	0.381396	−	0.924412i	$-0.375443\pi$
$5$	1.70566	0.762793	0.381396	+	0.924412i	$0.375443\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	8.02623	2.67541
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−3.32058	−0.920962	−0.460481	−	0.887669i	$-0.652323\pi$
$13$	−3.32058	−0.920962	−0.460481	+	0.887669i	$0.652323\pi$
$14$	0	0
$15$	5.66376	1.46238
$16$	0	0
$17$	0.867372	0.210369	0.105184	−	0.994453i	$-0.466457\pi$
$17$	0.867372	0.210369	0.105184	+	0.994453i	$0.466457\pi$
$18$	0	0
$19$	−2.27869	−0.522766	−0.261383	−	0.965235i	$-0.584179\pi$
$19$	−2.27869	−0.522766	−0.261383	+	0.965235i	$0.584179\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.61492	−0.545249	−0.272624	−	0.962121i	$-0.587892\pi$
$23$	−2.61492	−0.545249	−0.272624	+	0.962121i	$0.587892\pi$
$24$	0	0
$25$	−2.09074	−0.418147
$26$	0	0
$27$	16.6900	3.21199
$28$	0	0
$29$	9.34681	1.73566	0.867830	−	0.496862i	$-0.165514\pi$
$29$	9.34681	1.73566	0.867830	+	0.496862i	$0.165514\pi$
$30$	0	0
$31$	8.59926	1.54447	0.772237	−	0.635335i	$-0.219138\pi$
$31$	8.59926	1.54447	0.772237	+	0.635335i	$0.219138\pi$
$32$	0	0
$33$	3.32058	0.578038
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.41131	−1.21841	−0.609206	−	0.793012i	$-0.708512\pi$
$37$	−7.41131	−1.21841	−0.609206	+	0.793012i	$0.708512\pi$
$38$	0	0
$39$	−11.0262	−1.76561
$40$	0	0
$41$	4.83828	0.755613	0.377807	−	0.925885i	$-0.376678\pi$
$41$	4.83828	0.755613	0.377807	+	0.925885i	$0.376678\pi$
$42$	0	0
$43$	6.32058	0.963879	0.481940	−	0.876204i	$-0.339932\pi$
$43$	6.32058	0.963879	0.481940	+	0.876204i	$0.339932\pi$
$44$	0	0
$45$	13.6900	2.04078
$46$	0	0
$47$	2.40769	0.351198	0.175599	−	0.984462i	$-0.443814\pi$
$47$	2.40769	0.351198	0.175599	+	0.984462i	$0.443814\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	2.88018	0.403305
$52$	0	0
$53$	7.98434	1.09673	0.548367	−	0.836238i	$-0.315250\pi$
$53$	7.98434	1.09673	0.548367	+	0.836238i	$0.315250\pi$
$54$	0	0
$55$	1.70566	0.229991
$56$	0	0
$57$	−7.56655	−1.00221
$58$	0	0
$59$	11.0681	1.44095	0.720474	−	0.693482i	$-0.243924\pi$
$59$	11.0681	1.44095	0.720474	+	0.693482i	$0.243924\pi$
$60$	0	0
$61$	−7.43392	−0.951816	−0.475908	−	0.879495i	$-0.657881\pi$
$61$	−7.43392	−0.951816	−0.475908	+	0.879495i	$0.657881\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.66376	−0.702504
$66$	0	0
$67$	−6.48229	−0.791938	−0.395969	−	0.918264i	$-0.629591\pi$
$67$	−6.48229	−0.791938	−0.395969	+	0.918264i	$0.629591\pi$
$68$	0	0
$69$	−8.68305	−1.04532
$70$	0	0
$71$	−0.867372	−0.102938	−0.0514691	−	0.998675i	$-0.516390\pi$
$71$	−0.867372	−0.102938	−0.0514691	+	0.998675i	$0.516390\pi$
$72$	0	0
$73$	−13.6481	−1.59739	−0.798695	−	0.601736i	$-0.794476\pi$
$73$	−13.6481	−1.59739	−0.798695	+	0.601736i	$0.794476\pi$
$74$	0	0
$75$	−6.94245	−0.801645
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−9.47581	−1.06611	−0.533056	−	0.846080i	$-0.678957\pi$
$79$	−9.47581	−1.06611	−0.533056	+	0.846080i	$0.678957\pi$
$80$	0	0
$81$	31.3417	3.48241
$82$	0	0
$83$	−6.54394	−0.718291	−0.359145	−	0.933282i	$-0.616932\pi$
$83$	−6.54394	−0.718291	−0.359145	+	0.933282i	$0.616932\pi$
$84$	0	0
$85$	1.47944	0.160468
$86$	0	0
$87$	31.0368	3.32750
$88$	0	0
$89$	8.86452	0.939637	0.469819	−	0.882763i	$-0.344319\pi$
$89$	8.86452	0.939637	0.469819	+	0.882763i	$0.344319\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	28.5545	2.96097
$94$	0	0
$95$	−3.88665	−0.398762
$96$	0	0
$97$	13.2368	1.34399	0.671996	−	0.740554i	$-0.265437\pi$
$97$	13.2368	1.34399	0.671996	+	0.740554i	$0.265437\pi$
$98$	0	0
$99$	8.02623	0.806667
$100$	0	0
$101$	11.4149	1.13583	0.567914	−	0.823088i	$-0.307750\pi$
$101$	11.4149	1.13583	0.567914	+	0.823088i	$0.307750\pi$
$102$	0	0
$103$	−10.8489	−1.06897	−0.534485	−	0.845178i	$-0.679494\pi$
$103$	−10.8489	−1.06897	−0.534485	+	0.845178i	$0.679494\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.8517	1.04907	0.524537	−	0.851387i	$-0.324238\pi$
$107$	10.8517	1.04907	0.524537	+	0.851387i	$0.324238\pi$
$108$	0	0
$109$	−7.35662	−0.704636	−0.352318	−	0.935880i	$-0.614606\pi$
$109$	−7.35662	−0.704636	−0.352318	+	0.935880i	$0.614606\pi$
$110$	0	0
$111$	−24.6098	−2.33586
$112$	0	0
$113$	14.9424	1.40567	0.702834	−	0.711354i	$-0.251918\pi$
$113$	14.9424	1.40567	0.702834	+	0.711354i	$0.251918\pi$
$114$	0	0
$115$	−4.46016	−0.415912
$116$	0	0
$117$	−26.6517	−2.46395
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	16.0659	1.44861
$124$	0	0
$125$	−12.0944	−1.08175
$126$	0	0
$127$	4.80287	0.426186	0.213093	−	0.977032i	$-0.431646\pi$
$127$	4.80287	0.426186	0.213093	+	0.977032i	$0.431646\pi$
$128$	0	0
$129$	20.9880	1.84789
$130$	0	0
$131$	−6.86452	−0.599756	−0.299878	−	0.953978i	$-0.596946\pi$
$131$	−6.86452	−0.599756	−0.299878	+	0.953978i	$0.596946\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	28.4674	2.45008
$136$	0	0
$137$	10.9391	0.934592	0.467296	−	0.884101i	$-0.345228\pi$
$137$	10.9391	0.934592	0.467296	+	0.884101i	$0.345228\pi$
$138$	0	0
$139$	9.84524	0.835062	0.417531	−	0.908663i	$-0.362896\pi$
$139$	9.84524	0.835062	0.417531	+	0.908663i	$0.362896\pi$
$140$	0	0
$141$	7.99492	0.673294
$142$	0	0
$143$	−3.32058	−0.277681
$144$	0	0
$145$	15.9424	1.32395
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−10.8098	−0.885575	−0.442788	−	0.896627i	$-0.646010\pi$
$149$	−10.8098	−0.885575	−0.442788	+	0.896627i	$0.646010\pi$
$150$	0	0
$151$	2.67942	0.218048	0.109024	−	0.994039i	$-0.465227\pi$
$151$	2.67942	0.218048	0.109024	+	0.994039i	$0.465227\pi$
$152$	0	0
$153$	6.96173	0.562823
$154$	0	0
$155$	14.6674	1.17811
$156$	0	0
$157$	−20.3049	−1.62051	−0.810254	−	0.586078i	$-0.800671\pi$
$157$	−20.3049	−1.62051	−0.810254	+	0.586078i	$0.800671\pi$
$158$	0	0
$159$	26.5126	2.10259
$160$	0	0
$161$	0	0
$162$	0	0
$163$	23.2339	1.81982	0.909911	−	0.414803i	$-0.136149\pi$
$163$	23.2339	1.81982	0.909911	+	0.414803i	$0.136149\pi$
$164$	0	0
$165$	5.66376	0.440923
$166$	0	0
$167$	9.69585	0.750287	0.375144	−	0.926967i	$-0.377593\pi$
$167$	9.69585	0.750287	0.375144	+	0.926967i	$0.377593\pi$
$168$	0	0
$169$	−1.97377	−0.151828
$170$	0	0
$171$	−18.2893	−1.39861
$172$	0	0
$173$	−21.5020	−1.63477	−0.817385	−	0.576091i	$-0.804577\pi$
$173$	−21.5020	−1.63477	−0.817385	+	0.576091i	$0.804577\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	36.7526	2.76249
$178$	0	0
$179$	−2.47219	−0.184780	−0.0923901	−	0.995723i	$-0.529451\pi$
$179$	−2.47219	−0.184780	−0.0923901	+	0.995723i	$0.529451\pi$
$180$	0	0
$181$	9.04552	0.672348	0.336174	−	0.941800i	$-0.390867\pi$
$181$	9.04552	0.672348	0.336174	+	0.941800i	$0.390867\pi$
$182$	0	0
$183$	−24.6849	−1.82476
$184$	0	0
$185$	−12.6412	−0.929396
$186$	0	0
$187$	0.867372	0.0634285
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3.98720	0.288503	0.144252	−	0.989541i	$-0.453923\pi$
$191$	3.98720	0.288503	0.144252	+	0.989541i	$0.453923\pi$
$192$	0	0
$193$	−13.5319	−0.974048	−0.487024	−	0.873389i	$-0.661918\pi$
$193$	−13.5319	−0.974048	−0.487024	+	0.873389i	$0.661918\pi$
$194$	0	0
$195$	−18.8070	−1.34679
$196$	0	0
$197$	−6.45606	−0.459975	−0.229988	−	0.973194i	$-0.573869\pi$
$197$	−6.45606	−0.459975	−0.229988	+	0.973194i	$0.573869\pi$
$198$	0	0
$199$	−5.04599	−0.357701	−0.178850	−	0.983876i	$-0.557238\pi$
$199$	−5.04599	−0.357701	−0.178850	+	0.983876i	$0.557238\pi$
$200$	0	0
$201$	−21.5250	−1.51825
$202$	0	0
$203$	0	0
$204$	0	0
$205$	8.25245	0.576376
$206$	0	0
$207$	−20.9880	−1.45876
$208$	0	0
$209$	−2.27869	−0.157620
$210$	0	0
$211$	−2.18147	−0.150179	−0.0750893	−	0.997177i	$-0.523924\pi$
$211$	−2.18147	−0.150179	−0.0750893	+	0.997177i	$0.523924\pi$
$212$	0	0
$213$	−2.88018	−0.197346
$214$	0	0
$215$	10.7807	0.735240
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−45.3196	−3.06241
$220$	0	0
$221$	−2.88018	−0.193742
$222$	0	0
$223$	9.48229	0.634981	0.317491	−	0.948261i	$-0.397160\pi$
$223$	9.48229	0.634981	0.317491	+	0.948261i	$0.397160\pi$
$224$	0	0
$225$	−16.7807	−1.11872
$226$	0	0
$227$	26.3468	1.74870	0.874350	−	0.485297i	$-0.161288\pi$
$227$	26.3468	1.74870	0.874350	+	0.485297i	$0.161288\pi$
$228$	0	0
$229$	−14.2597	−0.942307	−0.471154	−	0.882051i	$-0.656162\pi$
$229$	−14.2597	−0.942307	−0.471154	+	0.882051i	$0.656162\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−13.7548	−0.901107	−0.450553	−	0.892750i	$-0.648773\pi$
$233$	−13.7548	−0.901107	−0.450553	+	0.892750i	$0.648773\pi$
$234$	0	0
$235$	4.10669	0.267891
$236$	0	0
$237$	−31.4652	−2.04388
$238$	0	0
$239$	6.92569	0.447986	0.223993	−	0.974591i	$-0.428091\pi$
$239$	6.92569	0.447986	0.223993	+	0.974591i	$0.428091\pi$
$240$	0	0
$241$	21.0907	1.35857	0.679287	−	0.733873i	$-0.262289\pi$
$241$	21.0907	1.35857	0.679287	+	0.733873i	$0.262289\pi$
$242$	0	0
$243$	54.0026	3.46427
$244$	0	0
$245$	0	0
$246$	0	0
$247$	7.56655	0.481448
$248$	0	0
$249$	−21.7297	−1.37706
$250$	0	0
$251$	0.996376	0.0628907	0.0314453	−	0.999505i	$-0.489989\pi$
$251$	0.996376	0.0628907	0.0314453	+	0.999505i	$0.489989\pi$
$252$	0	0
$253$	−2.61492	−0.164399
$254$	0	0
$255$	4.91259	0.307638
$256$	0	0
$257$	−24.5381	−1.53064	−0.765322	−	0.643648i	$-0.777420\pi$
$257$	−24.5381	−1.53064	−0.765322	+	0.643648i	$0.777420\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	75.0197	4.64360
$262$	0	0
$263$	−14.8263	−0.914226	−0.457113	−	0.889409i	$-0.651116\pi$
$263$	−14.8263	−0.914226	−0.457113	+	0.889409i	$0.651116\pi$
$264$	0	0
$265$	13.6185	0.836581
$266$	0	0
$267$	29.4353	1.80141
$268$	0	0
$269$	−7.95811	−0.485214	−0.242607	−	0.970125i	$-0.578003\pi$
$269$	−7.95811	−0.485214	−0.242607	+	0.970125i	$0.578003\pi$
$270$	0	0
$271$	−15.6098	−0.948230	−0.474115	−	0.880463i	$-0.657232\pi$
$271$	−15.6098	−0.948230	−0.474115	+	0.880463i	$0.657232\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.09074	−0.126076
$276$	0	0
$277$	−27.0441	−1.62492	−0.812460	−	0.583017i	$-0.801872\pi$
$277$	−27.0441	−1.62492	−0.812460	+	0.583017i	$0.801872\pi$
$278$	0	0
$279$	69.0197	4.13210
$280$	0	0
$281$	−19.6743	−1.17367	−0.586836	−	0.809706i	$-0.699627\pi$
$281$	−19.6743	−1.17367	−0.586836	+	0.809706i	$0.699627\pi$
$282$	0	0
$283$	−15.8972	−0.944992	−0.472496	−	0.881333i	$-0.656647\pi$
$283$	−15.8972	−0.944992	−0.472496	+	0.881333i	$0.656647\pi$
$284$	0	0
$285$	−12.9059	−0.764482
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.2477	−0.955745
$290$	0	0
$291$	43.9538	2.57662
$292$	0	0
$293$	−5.24265	−0.306279	−0.153139	−	0.988205i	$-0.548938\pi$
$293$	−5.24265	−0.306279	−0.153139	+	0.988205i	$0.548938\pi$
$294$	0	0
$295$	18.8784	1.09914
$296$	0	0
$297$	16.6900	0.968452
$298$	0	0
$299$	8.68305	0.502154
$300$	0	0
$301$	0	0
$302$	0	0
$303$	37.9042	2.17754
$304$	0	0
$305$	−12.6797	−0.726039
$306$	0	0
$307$	−14.0363	−0.801096	−0.400548	−	0.916276i	$-0.631180\pi$
$307$	−14.0363	−0.801096	−0.400548	+	0.916276i	$0.631180\pi$
$308$	0	0
$309$	−36.0245	−2.04936
$310$	0	0
$311$	13.0645	0.740820	0.370410	−	0.928868i	$-0.379217\pi$
$311$	13.0645	0.740820	0.370410	+	0.928868i	$0.379217\pi$
$312$	0	0
$313$	8.25608	0.466661	0.233330	−	0.972397i	$-0.425038\pi$
$313$	8.25608	0.466661	0.233330	+	0.972397i	$0.425038\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−11.0455	−0.620378	−0.310189	−	0.950675i	$-0.600392\pi$
$317$	−11.0455	−0.620378	−0.310189	+	0.950675i	$0.600392\pi$
$318$	0	0
$319$	9.34681	0.523321
$320$	0	0
$321$	36.0340	2.01122
$322$	0	0
$323$	−1.97647	−0.109974
$324$	0	0
$325$	6.94245	0.385098
$326$	0	0
$327$	−24.4282	−1.35088
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−25.8299	−1.41974	−0.709869	−	0.704334i	$-0.751246\pi$
$331$	−25.8299	−1.41974	−0.709869	+	0.704334i	$0.751246\pi$
$332$	0	0
$333$	−59.4849	−3.25975
$334$	0	0
$335$	−11.0566	−0.604085
$336$	0	0
$337$	−21.7425	−1.18439	−0.592194	−	0.805796i	$-0.701738\pi$
$337$	−21.7425	−1.18439	−0.592194	+	0.805796i	$0.701738\pi$
$338$	0	0
$339$	49.6176	2.69486
$340$	0	0
$341$	8.59926	0.465676
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−14.8103	−0.797360
$346$	0	0
$347$	1.27898	0.0686593	0.0343297	−	0.999411i	$-0.489070\pi$
$347$	1.27898	0.0686593	0.0343297	+	0.999411i	$0.489070\pi$
$348$	0	0
$349$	−26.6679	−1.42750	−0.713749	−	0.700402i	$-0.753004\pi$
$349$	−26.6679	−1.42750	−0.713749	+	0.700402i	$0.753004\pi$
$350$	0	0
$351$	−55.4204	−2.95812
$352$	0	0
$353$	−6.38585	−0.339884	−0.169942	−	0.985454i	$-0.554358\pi$
$353$	−6.38585	−0.339884	−0.169942	+	0.985454i	$0.554358\pi$
$354$	0	0
$355$	−1.47944	−0.0785205
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−0.185096	−0.00976897	−0.00488449	−	0.999988i	$-0.501555\pi$
$359$	−0.185096	−0.00976897	−0.00488449	+	0.999988i	$0.501555\pi$
$360$	0	0
$361$	−13.8076	−0.726715
$362$	0	0
$363$	3.32058	0.174285
$364$	0	0
$365$	−23.2790	−1.21848
$366$	0	0
$367$	−9.52528	−0.497216	−0.248608	−	0.968604i	$-0.579973\pi$
$367$	−9.52528	−0.497216	−0.248608	+	0.968604i	$0.579973\pi$
$368$	0	0
$369$	38.8332	2.02158
$370$	0	0
$371$	0	0
$372$	0	0
$373$	2.68228	0.138883	0.0694415	−	0.997586i	$-0.477878\pi$
$373$	2.68228	0.138883	0.0694415	+	0.997586i	$0.477878\pi$
$374$	0	0
$375$	−40.1603	−2.07387
$376$	0	0
$377$	−31.0368	−1.59848
$378$	0	0
$379$	1.02623	0.0527141	0.0263570	−	0.999653i	$-0.491609\pi$
$379$	1.02623	0.0527141	0.0263570	+	0.999653i	$0.491609\pi$
$380$	0	0
$381$	15.9483	0.817056
$382$	0	0
$383$	−9.80287	−0.500903	−0.250452	−	0.968129i	$-0.580579\pi$
$383$	−9.80287	−0.500903	−0.250452	+	0.968129i	$0.580579\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	50.7304	2.57877
$388$	0	0
$389$	32.9695	1.67162	0.835809	−	0.549021i	$-0.184999\pi$
$389$	32.9695	1.67162	0.835809	+	0.549021i	$0.184999\pi$
$390$	0	0
$391$	−2.26811	−0.114703
$392$	0	0
$393$	−22.7942	−1.14981
$394$	0	0
$395$	−16.1625	−0.813223
$396$	0	0
$397$	5.91622	0.296926	0.148463	−	0.988918i	$-0.452567\pi$
$397$	5.91622	0.296926	0.148463	+	0.988918i	$0.452567\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−14.7061	−0.734389	−0.367195	−	0.930144i	$-0.619682\pi$
$401$	−14.7061	−0.734389	−0.367195	+	0.930144i	$0.619682\pi$
$402$	0	0
$403$	−28.5545	−1.42240
$404$	0	0
$405$	53.4582	2.65636
$406$	0	0
$407$	−7.41131	−0.367365
$408$	0	0
$409$	10.1257	0.500682	0.250341	−	0.968158i	$-0.419457\pi$
$409$	10.1257	0.500682	0.250341	+	0.968158i	$0.419457\pi$
$410$	0	0
$411$	36.3242	1.79174
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−11.1617	−0.547907
$416$	0	0
$417$	32.6919	1.60093
$418$	0	0
$419$	−33.1348	−1.61874	−0.809370	−	0.587299i	$-0.800191\pi$
$419$	−33.1348	−1.61874	−0.809370	+	0.587299i	$0.800191\pi$
$420$	0	0
$421$	−28.9851	−1.41265	−0.706324	−	0.707889i	$-0.749648\pi$
$421$	−28.9851	−1.41265	−0.706324	+	0.707889i	$0.749648\pi$
$422$	0	0
$423$	19.3247	0.939598
$424$	0	0
$425$	−1.81345	−0.0879651
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−11.0262	−0.532352
$430$	0	0
$431$	8.19128	0.394560	0.197280	−	0.980347i	$-0.436789\pi$
$431$	8.19128	0.394560	0.197280	+	0.980347i	$0.436789\pi$
$432$	0	0
$433$	−8.42837	−0.405041	−0.202521	−	0.979278i	$-0.564913\pi$
$433$	−8.42837	−0.405041	−0.202521	+	0.979278i	$0.564913\pi$
$434$	0	0
$435$	52.9381	2.53819
$436$	0	0
$437$	5.95858	0.285038
$438$	0	0
$439$	23.5964	1.12620	0.563098	−	0.826390i	$-0.309610\pi$
$439$	23.5964	1.12620	0.563098	+	0.826390i	$0.309610\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	31.5001	1.49662	0.748308	−	0.663352i	$-0.230867\pi$
$443$	31.5001	1.49662	0.748308	+	0.663352i	$0.230867\pi$
$444$	0	0
$445$	15.1198	0.716748
$446$	0	0
$447$	−35.8948	−1.69777
$448$	0	0
$449$	19.7615	0.932601	0.466300	−	0.884626i	$-0.345587\pi$
$449$	19.7615	0.932601	0.466300	+	0.884626i	$0.345587\pi$
$450$	0	0
$451$	4.83828	0.227826
$452$	0	0
$453$	8.89723	0.418028
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−7.91003	−0.370016	−0.185008	−	0.982737i	$-0.559231\pi$
$457$	−7.91003	−0.370016	−0.185008	+	0.982737i	$0.559231\pi$
$458$	0	0
$459$	14.4764	0.675702
$460$	0	0
$461$	27.4094	1.27658	0.638291	−	0.769795i	$-0.279642\pi$
$461$	27.4094	1.27658	0.638291	+	0.769795i	$0.279642\pi$
$462$	0	0
$463$	7.80224	0.362601	0.181301	−	0.983428i	$-0.441969\pi$
$463$	7.80224	0.362601	0.181301	+	0.983428i	$0.441969\pi$
$464$	0	0
$465$	48.7042	2.25860
$466$	0	0
$467$	39.4973	1.82772	0.913858	−	0.406035i	$-0.133089\pi$
$467$	39.4973	1.82772	0.913858	+	0.406035i	$0.133089\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−67.4241	−3.10674
$472$	0	0
$473$	6.32058	0.290620
$474$	0	0
$475$	4.76413	0.218593
$476$	0	0
$477$	64.0842	2.93421
$478$	0	0
$479$	−2.80334	−0.128088	−0.0640440	−	0.997947i	$-0.520400\pi$
$479$	−2.80334	−0.128088	−0.0640440	+	0.997947i	$0.520400\pi$
$480$	0	0
$481$	24.6098	1.12211
$482$	0	0
$483$	0	0
$484$	0	0
$485$	22.5774	1.02519
$486$	0	0
$487$	4.36580	0.197833	0.0989166	−	0.995096i	$-0.468462\pi$
$487$	4.36580	0.197833	0.0989166	+	0.995096i	$0.468462\pi$
$488$	0	0
$489$	77.1501	3.48885
$490$	0	0
$491$	28.8493	1.30195	0.650976	−	0.759098i	$-0.274360\pi$
$491$	28.8493	1.30195	0.650976	+	0.759098i	$0.274360\pi$
$492$	0	0
$493$	8.10716	0.365128
$494$	0	0
$495$	13.6900	0.615320
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−2.83891	−0.127087	−0.0635435	−	0.997979i	$-0.520240\pi$
$499$	−2.83891	−0.127087	−0.0635435	+	0.997979i	$0.520240\pi$
$500$	0	0
$501$	32.1958	1.43840
$502$	0	0
$503$	−22.1701	−0.988514	−0.494257	−	0.869316i	$-0.664560\pi$
$503$	−22.1701	−0.988514	−0.494257	+	0.869316i	$0.664560\pi$
$504$	0	0
$505$	19.4700	0.866402
$506$	0	0
$507$	−6.55404	−0.291075
$508$	0	0
$509$	37.4448	1.65971	0.829856	−	0.557978i	$-0.188423\pi$
$509$	37.4448	1.65971	0.829856	+	0.557978i	$0.188423\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−38.0313	−1.67912
$514$	0	0
$515$	−18.5044	−0.815403
$516$	0	0
$517$	2.40769	0.105890
$518$	0	0
$519$	−71.3992	−3.13408
$520$	0	0
$521$	−14.2602	−0.624750	−0.312375	−	0.949959i	$-0.601124\pi$
$521$	−14.2602	−0.624750	−0.312375	+	0.949959i	$0.601124\pi$
$522$	0	0
$523$	5.47721	0.239502	0.119751	−	0.992804i	$-0.461790\pi$
$523$	5.47721	0.239502	0.119751	+	0.992804i	$0.461790\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.45876	0.324909
$528$	0	0
$529$	−16.1622	−0.702704
$530$	0	0
$531$	88.8354	3.85513
$532$	0	0
$533$	−16.0659	−0.695891
$534$	0	0
$535$	18.5093	0.800227
$536$	0	0
$537$	−8.20910	−0.354249
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−36.6051	−1.57377	−0.786887	−	0.617097i	$-0.788309\pi$
$541$	−36.6051	−1.57377	−0.786887	+	0.617097i	$0.788309\pi$
$542$	0	0
$543$	30.0363	1.28898
$544$	0	0
$545$	−12.5479	−0.537491
$546$	0	0
$547$	−7.42983	−0.317676	−0.158838	−	0.987305i	$-0.550775\pi$
$547$	−7.42983	−0.317676	−0.158838	+	0.987305i	$0.550775\pi$
$548$	0	0
$549$	−59.6664	−2.54650
$550$	0	0
$551$	−21.2984	−0.907344
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−41.9759	−1.78178
$556$	0	0
$557$	1.03319	0.0437775	0.0218887	−	0.999760i	$-0.493032\pi$
$557$	1.03319	0.0437775	0.0218887	+	0.999760i	$0.493032\pi$
$558$	0	0
$559$	−20.9880	−0.887696
$560$	0	0
$561$	2.88018	0.121601
$562$	0	0
$563$	−10.0234	−0.422435	−0.211218	−	0.977439i	$-0.567743\pi$
$563$	−10.0234	−0.422435	−0.211218	+	0.977439i	$0.567743\pi$
$564$	0	0
$565$	25.4867	1.07223
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−19.8134	−0.830623	−0.415311	−	0.909679i	$-0.636327\pi$
$569$	−19.8134	−0.830623	−0.415311	+	0.909679i	$0.636327\pi$
$570$	0	0
$571$	−28.4886	−1.19221	−0.596104	−	0.802907i	$-0.703286\pi$
$571$	−28.4886	−1.19221	−0.596104	+	0.802907i	$0.703286\pi$
$572$	0	0
$573$	13.2398	0.553100
$574$	0	0
$575$	5.46711	0.227994
$576$	0	0
$577$	−11.5384	−0.480349	−0.240175	−	0.970730i	$-0.577205\pi$
$577$	−11.5384	−0.480349	−0.240175	+	0.970730i	$0.577205\pi$
$578$	0	0
$579$	−44.9337	−1.86738
$580$	0	0
$581$	0	0
$582$	0	0
$583$	7.98434	0.330678
$584$	0	0
$585$	−45.4587	−1.87949
$586$	0	0
$587$	−7.07537	−0.292032	−0.146016	−	0.989282i	$-0.546645\pi$
$587$	−7.07537	−0.292032	−0.146016	+	0.989282i	$0.546645\pi$
$588$	0	0
$589$	−19.5950	−0.807398
$590$	0	0
$591$	−21.4378	−0.881835
$592$	0	0
$593$	18.5098	0.760105	0.380053	−	0.924965i	$-0.375906\pi$
$593$	18.5098	0.760105	0.380053	+	0.924965i	$0.375906\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−16.7556	−0.685761
$598$	0	0
$599$	−15.1562	−0.619264	−0.309632	−	0.950856i	$-0.600206\pi$
$599$	−15.1562	−0.619264	−0.309632	+	0.950856i	$0.600206\pi$
$600$	0	0
$601$	12.7540	0.520248	0.260124	−	0.965575i	$-0.416237\pi$
$601$	12.7540	0.520248	0.260124	+	0.965575i	$0.416237\pi$
$602$	0	0
$603$	−52.0284	−2.11876
$604$	0	0
$605$	1.70566	0.0693448
$606$	0	0
$607$	−29.7788	−1.20868	−0.604342	−	0.796725i	$-0.706564\pi$
$607$	−29.7788	−1.20868	−0.604342	+	0.796725i	$0.706564\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−7.99492	−0.323440
$612$	0	0
$613$	−5.27506	−0.213058	−0.106529	−	0.994310i	$-0.533974\pi$
$613$	−5.27506	−0.213058	−0.106529	+	0.994310i	$0.533974\pi$
$614$	0	0
$615$	27.4029	1.10499
$616$	0	0
$617$	25.8411	1.04033	0.520163	−	0.854067i	$-0.325871\pi$
$617$	25.8411	1.04033	0.520163	+	0.854067i	$0.325871\pi$
$618$	0	0
$619$	4.22669	0.169885	0.0849425	−	0.996386i	$-0.472929\pi$
$619$	4.22669	0.169885	0.0849425	+	0.996386i	$0.472929\pi$
$620$	0	0
$621$	−43.6430	−1.75133
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−10.1751	−0.407006
$626$	0	0
$627$	−7.56655	−0.302179
$628$	0	0
$629$	−6.42837	−0.256316
$630$	0	0
$631$	20.8779	0.831138	0.415569	−	0.909562i	$-0.363582\pi$
$631$	20.8779	0.831138	0.415569	+	0.909562i	$0.363582\pi$
$632$	0	0
$633$	−7.24374	−0.287913
$634$	0	0
$635$	8.19205	0.325091
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−6.96173	−0.275402
$640$	0	0
$641$	−42.4476	−1.67658	−0.838291	−	0.545224i	$-0.816445\pi$
$641$	−42.4476	−1.67658	−0.838291	+	0.545224i	$0.816445\pi$
$642$	0	0
$643$	35.7064	1.40812	0.704062	−	0.710139i	$-0.251368\pi$
$643$	35.7064	1.40812	0.704062	+	0.710139i	$0.251368\pi$
$644$	0	0
$645$	35.7983	1.40956
$646$	0	0
$647$	−2.31048	−0.0908342	−0.0454171	−	0.998968i	$-0.514462\pi$
$647$	−2.31048	−0.0908342	−0.0454171	+	0.998968i	$0.514462\pi$
$648$	0	0
$649$	11.0681	0.434462
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3.19398	0.124990	0.0624950	−	0.998045i	$-0.480094\pi$
$653$	3.19398	0.124990	0.0624950	+	0.998045i	$0.480094\pi$
$654$	0	0
$655$	−11.7085	−0.457489
$656$	0	0
$657$	−109.543	−4.27368
$658$	0	0
$659$	−7.94970	−0.309676	−0.154838	−	0.987940i	$-0.549486\pi$
$659$	−7.94970	−0.309676	−0.154838	+	0.987940i	$0.549486\pi$
$660$	0	0
$661$	−17.0790	−0.664296	−0.332148	−	0.943227i	$-0.607773\pi$
$661$	−17.0790	−0.664296	−0.332148	+	0.943227i	$0.607773\pi$
$662$	0	0
$663$	−9.56385	−0.371429
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−24.4412	−0.946366
$668$	0	0
$669$	31.4867	1.21735
$670$	0	0
$671$	−7.43392	−0.286983
$672$	0	0
$673$	−43.7770	−1.68748	−0.843741	−	0.536751i	$-0.819651\pi$
$673$	−43.7770	−1.68748	−0.843741	+	0.536751i	$0.819651\pi$
$674$	0	0
$675$	−34.8944	−1.34309
$676$	0	0
$677$	4.74562	0.182389	0.0911944	−	0.995833i	$-0.470932\pi$
$677$	4.74562	0.182389	0.0911944	+	0.995833i	$0.470932\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	87.4866	3.35249
$682$	0	0
$683$	47.5814	1.82065	0.910325	−	0.413893i	$-0.135831\pi$
$683$	47.5814	1.82065	0.910325	+	0.413893i	$0.135831\pi$
$684$	0	0
$685$	18.6584	0.712900
$686$	0	0
$687$	−47.3504	−1.80653
$688$	0	0
$689$	−26.5126	−1.01005
$690$	0	0
$691$	33.5657	1.27690	0.638450	−	0.769663i	$-0.279576\pi$
$691$	33.5657	1.27690	0.638450	+	0.769663i	$0.279576\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	16.7926	0.636979
$696$	0	0
$697$	4.19659	0.158957
$698$	0	0
$699$	−45.6739	−1.72754
$700$	0	0
$701$	−1.34967	−0.0509761	−0.0254881	−	0.999675i	$-0.508114\pi$
$701$	−1.34967	−0.0509761	−0.0254881	+	0.999675i	$0.508114\pi$
$702$	0	0
$703$	16.8881	0.636945
$704$	0	0
$705$	13.6366	0.513583
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−45.0671	−1.69253	−0.846266	−	0.532761i	$-0.821155\pi$
$709$	−45.0671	−1.69253	−0.846266	+	0.532761i	$0.821155\pi$
$710$	0	0
$711$	−76.0551	−2.85229
$712$	0	0
$713$	−22.4864	−0.842122
$714$	0	0
$715$	−5.66376	−0.211813
$716$	0	0
$717$	22.9973	0.858850
$718$	0	0
$719$	9.25322	0.345087	0.172543	−	0.985002i	$-0.444801\pi$
$719$	9.25322	0.345087	0.172543	+	0.985002i	$0.444801\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	70.0334	2.60457
$724$	0	0
$725$	−19.5417	−0.725761
$726$	0	0
$727$	25.1437	0.932527	0.466264	−	0.884646i	$-0.345600\pi$
$727$	25.1437	0.932527	0.466264	+	0.884646i	$0.345600\pi$
$728$	0	0
$729$	85.2948	3.15906
$730$	0	0
$731$	5.48229	0.202770
$732$	0	0
$733$	32.7749	1.21057	0.605284	−	0.796010i	$-0.293060\pi$
$733$	32.7749	1.21057	0.605284	+	0.796010i	$0.293060\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.48229	−0.238778
$738$	0	0
$739$	52.3360	1.92521	0.962606	−	0.270906i	$-0.0873232\pi$
$739$	52.3360	1.92521	0.962606	+	0.270906i	$0.0873232\pi$
$740$	0	0
$741$	25.1253	0.923002
$742$	0	0
$743$	−37.0985	−1.36101	−0.680505	−	0.732743i	$-0.738240\pi$
$743$	−37.0985	−1.36101	−0.680505	+	0.732743i	$0.738240\pi$
$744$	0	0
$745$	−18.4378	−0.675510
$746$	0	0
$747$	−52.5232	−1.92172
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−51.2219	−1.86911	−0.934557	−	0.355814i	$-0.884204\pi$
$751$	−51.2219	−1.86911	−0.934557	+	0.355814i	$0.884204\pi$
$752$	0	0
$753$	3.30854	0.120570
$754$	0	0
$755$	4.57017	0.166326
$756$	0	0
$757$	−27.1299	−0.986054	−0.493027	−	0.870014i	$-0.664110\pi$
$757$	−27.1299	−0.986054	−0.493027	+	0.870014i	$0.664110\pi$
$758$	0	0
$759$	−8.68305	−0.315175
$760$	0	0
$761$	−26.0022	−0.942578	−0.471289	−	0.881979i	$-0.656211\pi$
$761$	−26.0022	−0.942578	−0.471289	+	0.881979i	$0.656211\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	11.8743	0.429317
$766$	0	0
$767$	−36.7526	−1.32706
$768$	0	0
$769$	31.0365	1.11921	0.559603	−	0.828761i	$-0.310954\pi$
$769$	31.0365	1.11921	0.559603	+	0.828761i	$0.310954\pi$
$770$	0	0
$771$	−81.4806	−2.93445
$772$	0	0
$773$	21.1538	0.760849	0.380424	−	0.924812i	$-0.375778\pi$
$773$	21.1538	0.760849	0.380424	+	0.924812i	$0.375778\pi$
$774$	0	0
$775$	−17.9788	−0.645817
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−11.0249	−0.395009
$780$	0	0
$781$	−0.867372	−0.0310370
$782$	0	0
$783$	155.998	5.57492
$784$	0	0
$785$	−34.6332	−1.23611
$786$	0	0
$787$	−38.7617	−1.38171	−0.690854	−	0.722995i	$-0.742765\pi$
$787$	−38.7617	−1.38171	−0.690854	+	0.722995i	$0.742765\pi$
$788$	0	0
$789$	−49.2317	−1.75270
$790$	0	0
$791$	0	0
$792$	0	0
$793$	24.6849	0.876587
$794$	0	0
$795$	45.2214	1.60384
$796$	0	0
$797$	−22.9384	−0.812518	−0.406259	−	0.913758i	$-0.633167\pi$
$797$	−22.9384	−0.812518	−0.406259	+	0.913758i	$0.633167\pi$
$798$	0	0
$799$	2.08836	0.0738810
$800$	0	0
$801$	71.1487	2.51392
$802$	0	0
$803$	−13.6481	−0.481631
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−26.4255	−0.930222
$808$	0	0
$809$	52.2992	1.83874	0.919371	−	0.393391i	$-0.128698\pi$
$809$	52.2992	1.83874	0.919371	+	0.393391i	$0.128698\pi$
$810$	0	0
$811$	−10.5808	−0.371540	−0.185770	−	0.982593i	$-0.559478\pi$
$811$	−10.5808	−0.371540	−0.185770	+	0.982593i	$0.559478\pi$
$812$	0	0
$813$	−51.8337	−1.81789
$814$	0	0
$815$	39.6291	1.38815
$816$	0	0
$817$	−14.4026	−0.503883
$818$	0	0
$819$	0	0
$820$	0	0
$821$	12.5581	0.438282	0.219141	−	0.975693i	$-0.429675\pi$
$821$	12.5581	0.438282	0.219141	+	0.975693i	$0.429675\pi$
$822$	0	0
$823$	−10.2306	−0.356617	−0.178308	−	0.983975i	$-0.557062\pi$
$823$	−10.2306	−0.356617	−0.178308	+	0.983975i	$0.557062\pi$
$824$	0	0
$825$	−6.94245	−0.241705
$826$	0	0
$827$	−38.3160	−1.33238	−0.666188	−	0.745784i	$-0.732075\pi$
$827$	−38.3160	−1.33238	−0.666188	+	0.745784i	$0.732075\pi$
$828$	0	0
$829$	−1.52481	−0.0529589	−0.0264794	−	0.999649i	$-0.508430\pi$
$829$	−1.52481	−0.0529589	−0.0264794	+	0.999649i	$0.508430\pi$
$830$	0	0
$831$	−89.8019	−3.11519
$832$	0	0
$833$	0	0
$834$	0	0
$835$	16.5378	0.572314
$836$	0	0
$837$	143.522	4.96083
$838$	0	0
$839$	2.04204	0.0704992	0.0352496	−	0.999379i	$-0.488777\pi$
$839$	2.04204	0.0704992	0.0352496	+	0.999379i	$0.488777\pi$
$840$	0	0
$841$	58.3629	2.01251
$842$	0	0
$843$	−65.3302	−2.25009
$844$	0	0
$845$	−3.36657	−0.115813
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−52.7880	−1.81168
$850$	0	0
$851$	19.3800	0.664338
$852$	0	0
$853$	15.1192	0.517671	0.258836	−	0.965921i	$-0.416661\pi$
$853$	15.1192	0.517671	0.258836	+	0.965921i	$0.416661\pi$
$854$	0	0
$855$	−31.1952	−1.06685
$856$	0	0
$857$	−28.5640	−0.975727	−0.487864	−	0.872920i	$-0.662224\pi$
$857$	−28.5640	−0.975727	−0.487864	+	0.872920i	$0.662224\pi$
$858$	0	0
$859$	33.7037	1.14996	0.574978	−	0.818169i	$-0.305011\pi$
$859$	33.7037	1.14996	0.574978	+	0.818169i	$0.305011\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−21.4929	−0.731626	−0.365813	−	0.930688i	$-0.619209\pi$
$863$	−21.4929	−0.731626	−0.365813	+	0.930688i	$0.619209\pi$
$864$	0	0
$865$	−36.6751	−1.24699
$866$	0	0
$867$	−53.9516	−1.83229
$868$	0	0
$869$	−9.47581	−0.321445
$870$	0	0
$871$	21.5250	0.729345
$872$	0	0
$873$	106.242	3.59573
$874$	0	0
$875$	0	0
$876$	0	0
$877$	4.13040	0.139474	0.0697368	−	0.997565i	$-0.477784\pi$
$877$	4.13040	0.139474	0.0697368	+	0.997565i	$0.477784\pi$
$878$	0	0
$879$	−17.4086	−0.587178
$880$	0	0
$881$	−25.0692	−0.844604	−0.422302	−	0.906455i	$-0.638778\pi$
$881$	−25.0692	−0.844604	−0.422302	+	0.906455i	$0.638778\pi$
$882$	0	0
$883$	−25.6579	−0.863457	−0.431729	−	0.902003i	$-0.642096\pi$
$883$	−25.6579	−0.863457	−0.431729	+	0.902003i	$0.642096\pi$
$884$	0	0
$885$	62.6873	2.10721
$886$	0	0
$887$	2.37783	0.0798398	0.0399199	−	0.999203i	$-0.487290\pi$
$887$	2.37783	0.0798398	0.0399199	+	0.999203i	$0.487290\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	31.3417	1.04999
$892$	0	0
$893$	−5.48637	−0.183594
$894$	0	0
$895$	−4.21671	−0.140949
$896$	0	0
$897$	28.8327	0.962697
$898$	0	0
$899$	80.3757	2.68068
$900$	0	0
$901$	6.92540	0.230718
$902$	0	0
$903$	0	0
$904$	0	0
$905$	15.4285	0.512862
$906$	0	0
$907$	−15.8212	−0.525333	−0.262667	−	0.964887i	$-0.584602\pi$
$907$	−15.8212	−0.525333	−0.262667	+	0.964887i	$0.584602\pi$
$908$	0	0
$909$	91.6190	3.03881
$910$	0	0
$911$	32.4388	1.07475	0.537373	−	0.843345i	$-0.319417\pi$
$911$	32.4388	1.07475	0.537373	+	0.843345i	$0.319417\pi$
$912$	0	0
$913$	−6.54394	−0.216573
$914$	0	0
$915$	−42.1040	−1.39191
$916$	0	0
$917$	0	0
$918$	0	0
$919$	58.5913	1.93275	0.966375	−	0.257138i	$-0.0827794\pi$
$919$	58.5913	1.93275	0.966375	+	0.257138i	$0.0827794\pi$
$920$	0	0
$921$	−46.6087	−1.53581
$922$	0	0
$923$	2.88018	0.0948022
$924$	0	0
$925$	15.4951	0.509476
$926$	0	0
$927$	−87.0755	−2.85993
$928$	0	0
$929$	23.9366	0.785334	0.392667	−	0.919681i	$-0.371552\pi$
$929$	23.9366	0.785334	0.392667	+	0.919681i	$0.371552\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	43.3817	1.42025
$934$	0	0
$935$	1.47944	0.0483828
$936$	0	0
$937$	−22.3735	−0.730911	−0.365455	−	0.930829i	$-0.619087\pi$
$937$	−22.3735	−0.730911	−0.365455	+	0.930829i	$0.619087\pi$
$938$	0	0
$939$	27.4149	0.894653
$940$	0	0
$941$	0.737446	0.0240401	0.0120200	−	0.999928i	$-0.496174\pi$
$941$	0.737446	0.0240401	0.0120200	+	0.999928i	$0.496174\pi$
$942$	0	0
$943$	−12.6517	−0.411997
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.99052	0.129675	0.0648373	−	0.997896i	$-0.479347\pi$
$947$	3.99052	0.129675	0.0648373	+	0.997896i	$0.479347\pi$
$948$	0	0
$949$	45.3196	1.47114
$950$	0	0
$951$	−36.6775	−1.18935
$952$	0	0
$953$	3.16504	0.102526	0.0512629	−	0.998685i	$-0.483675\pi$
$953$	3.16504	0.102526	0.0512629	+	0.998685i	$0.483675\pi$
$954$	0	0
$955$	6.80079	0.220068
$956$	0	0
$957$	31.0368	1.00328
$958$	0	0
$959$	0	0
$960$	0	0
$961$	42.9473	1.38540
$962$	0	0
$963$	87.0984	2.80671
$964$	0	0
$965$	−23.0808	−0.742997
$966$	0	0
$967$	−42.3673	−1.36244	−0.681221	−	0.732078i	$-0.738551\pi$
$967$	−42.3673	−1.36244	−0.681221	+	0.732078i	$0.738551\pi$
$968$	0	0
$969$	−6.56302	−0.210834
$970$	0	0
$971$	4.02769	0.129255	0.0646274	−	0.997909i	$-0.479414\pi$
$971$	4.02769	0.129255	0.0646274	+	0.997909i	$0.479414\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	23.0529	0.738285
$976$	0	0
$977$	−40.7979	−1.30524	−0.652621	−	0.757685i	$-0.726330\pi$
$977$	−40.7979	−1.30524	−0.652621	+	0.757685i	$0.726330\pi$
$978$	0	0
$979$	8.86452	0.283311
$980$	0	0
$981$	−59.0459	−1.88519
$982$	0	0
$983$	29.0209	0.925622	0.462811	−	0.886457i	$-0.346841\pi$
$983$	29.0209	0.925622	0.462811	+	0.886457i	$0.346841\pi$
$984$	0	0
$985$	−11.0118	−0.350866
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−16.5278	−0.525554
$990$	0	0
$991$	30.6031	0.972138	0.486069	−	0.873920i	$-0.338430\pi$
$991$	30.6031	0.972138	0.486069	+	0.873920i	$0.338430\pi$
$992$	0	0
$993$	−85.7701	−2.72183
$994$	0	0
$995$	−8.60673	−0.272852
$996$	0	0
$997$	44.0112	1.39385	0.696924	−	0.717145i	$-0.254551\pi$
$997$	44.0112	1.39385	0.696924	+	0.717145i	$0.254551\pi$
$998$	0	0
$999$	−123.695	−3.91353

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.cz.1.4		4
4.3	odd	2		4312.2.a.y.1.1		4
7.2	even	3		1232.2.q.n.529.1		8
7.4	even	3		1232.2.q.n.177.1		8
7.6	odd	2		8624.2.a.cr.1.1		4
28.11	odd	6		616.2.q.d.177.4	✓	8
28.23	odd	6		616.2.q.d.529.4	yes	8
28.27	even	2		4312.2.a.bd.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.2.q.d.177.4	✓	8	28.11	odd	6
616.2.q.d.529.4	yes	8	28.23	odd	6
1232.2.q.n.177.1		8	7.4	even	3
1232.2.q.n.529.1		8	7.2	even	3
4312.2.a.y.1.1		4	4.3	odd	2
4312.2.a.bd.1.4		4	28.27	even	2
8624.2.a.cr.1.1		4	7.6	odd	2
8624.2.a.cz.1.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8624.a (trivial)

\(f(q)\)	\(=\)	\(q+3.32058 q^{3} +1.70566 q^{5} +8.02623 q^{9} +O(q^{10})\)	\(q+3.32058 q^{3} +1.70566 q^{5} +8.02623 q^{9} +1.00000 q^{11} -3.32058 q^{13} +5.66376 q^{15} +0.867372 q^{17} -2.27869 q^{19} -2.61492 q^{23} -2.09074 q^{25} +16.6900 q^{27} +9.34681 q^{29} +8.59926 q^{31} +3.32058 q^{33} -7.41131 q^{37} -11.0262 q^{39} +4.83828 q^{41} +6.32058 q^{43} +13.6900 q^{45} +2.40769 q^{47} +2.88018 q^{51} +7.98434 q^{53} +1.70566 q^{55} -7.56655 q^{57} +11.0681 q^{59} -7.43392 q^{61} -5.66376 q^{65} -6.48229 q^{67} -8.68305 q^{69} -0.867372 q^{71} -13.6481 q^{73} -6.94245 q^{75} -9.47581 q^{79} +31.3417 q^{81} -6.54394 q^{83} +1.47944 q^{85} +31.0368 q^{87} +8.86452 q^{89} +28.5545 q^{93} -3.88665 q^{95} +13.2368 q^{97} +8.02623 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 4 q^{5} + 10 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 - 4 * q^5 + 10 * q^9	\( 4 q + 2 q^{3} - 4 q^{5} + 10 q^{9} + 4 q^{11} - 2 q^{13} + q^{15} + 3 q^{17} + 13 q^{19} - 10 q^{23} + 2 q^{25} + 23 q^{27} + 4 q^{29} + q^{31} + 2 q^{33} - 8 q^{37} - 22 q^{39} + 9 q^{41} + 14 q^{43} + 11 q^{45} + 11 q^{47} + 12 q^{51} - q^{53} - 4 q^{55} - 10 q^{57} + 33 q^{59} - 9 q^{61} - q^{65} - 25 q^{67} - 23 q^{69} - 3 q^{71} + 16 q^{75} - 28 q^{79} + 4 q^{81} - 5 q^{83} - 27 q^{85} + 47 q^{87} + 3 q^{89} + 38 q^{93} - 25 q^{95} + 20 q^{97} + 10 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 - 4 * q^5 + 10 * q^9 + 4 * q^11 - 2 * q^13 + q^15 + 3 * q^17 + 13 * q^19 - 10 * q^23 + 2 * q^25 + 23 * q^27 + 4 * q^29 + q^31 + 2 * q^33 - 8 * q^37 - 22 * q^39 + 9 * q^41 + 14 * q^43 + 11 * q^45 + 11 * q^47 + 12 * q^51 - q^53 - 4 * q^55 - 10 * q^57 + 33 * q^59 - 9 * q^61 - q^65 - 25 * q^67 - 23 * q^69 - 3 * q^71 + 16 * q^75 - 28 * q^79 + 4 * q^81 - 5 * q^83 - 27 * q^85 + 47 * q^87 + 3 * q^89 + 38 * q^93 - 25 * q^95 + 20 * q^97 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more